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</o:shapelayout></xml><![endif]--></head><body lang=EN-US link="#0563C1" vlink="#954F72"><div class=WordSection1><p class=MsoNormal><span style='color:black'>Hi Iman,<o:p></o:p></span></p><p class=MsoNormal><span style='color:black'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:black'>The reference used in average referencing, or single channel or avg fiducials reference, is a linear function of the data. So e.g. in avg reference,<o:p></o:p></span></p><p class=MsoNormal><span style='color:black'><o:p> </o:p></span></p><p class=MsoNormal style='text-indent:.5in'><span style='color:black'>r(t) = e^T * x(t) / n<o:p></o:p></span></p><p class=MsoNormal><span style='color:black'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:black'>where e is a vector of all ones, e = [1 1 1 … ]^T, and ^T means transpose, and n is number of channels.<o:p></o:p></span></p><p class=MsoNormal><span style='color:black'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:black'>Re-referencing the data is equivalent to doing:<o:p></o:p></span></p><p class=MsoNormal><span style='color:black'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:black'> y(t) = x(t) – e*e^T*x(t) / n<o:p></o:p></span></p><p class=MsoNormal style='text-indent:.5in'><span style='color:black'> = (I – e*e^T/n) * x(t)<o:p></o:p></span></p><p class=MsoNormal><span style='color:black'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:black'>Where I is the identity matrix, which shows that the re-referenced data is just a matrix times the original data, and has rank reduced by 1. Similarly for avgs of subsets of channels, which use I – e*g^T with a general vector g, instead of I – e*e^T.<o:p></o:p></span></p><p class=MsoNormal><span style='color:black'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:black'>So we can look at the new data as generated by,<o:p></o:p></span></p><p class=MsoNormal><span style='color:black'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:black'> </span><span lang=FR style='color:black'>y(t) = M’ * s(t)<o:p></o:p></span></p><p class=MsoNormal><span lang=FR style='color:black'><o:p> </o:p></span></p><p class=MsoNormal><span lang=FR style='color:black'>where,<o:p></o:p></span></p><p class=MsoNormal><span lang=FR style='color:black'><o:p> </o:p></span></p><p class=MsoNormal style='text-indent:.5in'><span lang=IT style='color:black'>M’ = (I – e*g^T)*M<o:p></o:p></span></p><p class=MsoNormal style='text-indent:.5in'><span lang=IT style='color:black'> = M – e*(g^T*M)<o:p></o:p></span></p><p class=MsoNormal><span lang=IT style='color:black'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:black'>This new mixing matrix is just the original matrix with a single constant potential subtracted from each mixing map (generally different number for each map, but same number for each channel of a map).<o:p></o:p></span></p><p class=MsoNormal><span style='color:black'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:black'>So what the reference does is essentially add a constant to each channel. Average reference will tend to produce maps such that the sum over the channels of the potential of each map is zero, consistent with charge conservation and dipolarity.<o:p></o:p></span></p><p class=MsoNormal><span style='color:black'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:black'>So when you do dipole localization with a given map, there is really an arbitrary additive constant, but presumably only one value of this constant will be consistent with a physiological source.<o:p></o:p></span></p><p class=MsoNormal><span style='color:black'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:black'>Hope that is helpful.<o:p></o:p></span></p><p class=MsoNormal><span style='color:black'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:black'>Best,<o:p></o:p></span></p><p class=MsoNormal><span style='color:black'>Jason<o:p></o:p></span></p><p class=MsoNormal><span style='color:black'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:black'><o:p> </o:p></span></p><div><div style='border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0in 0in 0in'><p class=MsoNormal><b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>From:</span></b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'> eeglablist-bounces@sccn.ucsd.edu [mailto:eeglablist-bounces@sccn.ucsd.edu] <b>On Behalf Of </b>Iman M.Rezazadeh<br><b>Sent:</b> Monday, March 17, 2014 4:31 PM<br><b>To:</b> 'EEGLAB List'<br><b>Subject:</b> [Eeglablist] why ICA is reference-free?<o:p></o:p></span></p></div></div><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Hi, <o:p></o:p></p><p class=MsoNormal>Could you please elaborate why ICA is reference-free? In other words, does different kinds of referencing methods (avg-ref or …) effect on ICA maps and source series ? <o:p></o:p></p><p class=MsoNormal>Suppose: x(t) is the signal then we have s(t)=M(^-1). x(t) ;where s(t) is source signals and M is the mixing matrix.<o:p></o:p></p><p class=MsoNormal>Now how come for y=x(t)-r(t) - where r(t) is another reference- result same/different maps and source series would be obtained and how could someone get the same dipole locations with different re-referencing methods?<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>It would be great if someone gives in depth <b>methodological and mathematical explanations</b> rather than just qualitative explanations.<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Best<o:p></o:p></p><p class=MsoNormal>Iman<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><b>-------------------------------------------------------------<o:p></o:p></b></p><p class=MsoNormal><b>Iman M.Rezazadeh, Ph.D<o:p></o:p></b></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Research Fellow<o:p></o:p></p><p class=MsoNormal>Semel Intitute, UCLA , Los Angeles<o:p></o:p></p><p class=MsoNormal>& Center for Mind and Brain, UC DAVIS, Davis<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><o:p> </o:p></p></div></body></html>